The Action ofF4(q) on Cosets ofB4(q)

In this paper we consider the action of the simple groupF4(q) on the cosets of the maximal subgroupB4(q). We show that the action is multiplicity-free of rankq + 3; we obtain suborbit representatives and calculate subdegrees, show that all suborbits are self-paired, find that none of the graphs arising from the action is distance-transitive, and give explicitly the decomposition of the permutation character. In addition, we give detailed information on the correspondence between geometric conjugacy classes and semisimple classes which is used in the Deligne–Lusztig theory.